14.452 Economic Growth: Lecture 13, Directed Technological Change

14.452 Economic Growth: Lecture 13, DirectedTechnological Change

Daron Acemoglu

MIT

December 13, 2011.

Daron Acemoglu (MIT) Economic Growth Lecture 13 December 13, 2011. 1 / 71

Directed Technological Change Introduction

Introduction

Thus far have focused on a single type of technological change (e.g.,Hicks-neutral).

But, technological change is often not neutral:1 Benefits some factors of production and some agents more than others.Distributional effects imply some groups will embrace new technologiesand others oppose them.

2 Limiting to only one type of technological change obscures thecompeting effects that determine the nature of technological change.

Directed technological change: endogenize the direction and bias ofnew technologies that are developed and adopted.


Biased Technological Change Importance

Skill-biased technological change

Over the past 60 years, the U.S. relative supply of skills has increased,but:

1 there has also been an increase in the college premium, and2 this increase accelerated in the late 1960s, and the skill premiumincreased very rapidly beginning in the late 1970s.

Standard explanation: skill bias technical change, and an accelerationthat coincided with the changes in the relative supply of skills.

Important question: skill bias is endogenous, so, why hastechnological change become more skill biased in recent decades?



Skill-biased technological changeC

olle

ge w

age

prem

ium

Relative Supply of College Skills and College Premiumyear

Rel

. su

pply

of

colle

ge s

kills

College wage premium Rel. supply of college skills

39 49 59 69 79 89 96.3

.4

.5

.6

0

.2

.4

.6

.8

Figure:Daron Acemoglu (MIT) Economic Growth Lecture 13 December 13, 2011. 4 / 71


Unskill-biased technological change

Late 18th and early 19th unskill-bias:“First in firearms, then in clocks, pumps, locks, mechanical reapers,typewriters, sewing machines, and eventually in engines and bicycles,interchangeable parts technology proved superior and replaced theskilled artisans working with chisel and file.” (Mokyr 1990, p. 137)

Why was technological change unskilled-biased then andskilled-biased now?



Wage push and capital-biased technological change

First phase. Late 1960s and early 1970s: unemployment and share oflabor in national income increased rapidly continental Europeancountries.

Second phase. 1980s: unemployment continued to increase, but thelabor share declined, even below its initial level.

Blanchard (1997):

Phase 1: wage-push by workersPhase 2: capital-biased technological changes.

Is there a connection between capital-biased technological changes inEuropean economies and the wage push preceding it?



Importance of Biased Technological Change: moreexamples

Balanced economic growth:

Only possible when technological change is asymptoticallyHarrod-neutral, i.e., purely labor augmenting.Is there any reason to expect technological change to be endogenouslylabor augmenting?

Globalization:

Does it affect the types of technologies that are being developed andused?



Directed Technological Change: Basic Arguments I

Two factors of production, say L and H (unskilled and skilledworkers).

Two types of technologies that can complement either one or theother factor.

Whenever the profitability of H-augmenting technologies is greaterthan the L-augmenting technologies, more of the former type will bedeveloped by profit-maximizing (research) firms.

What determines the relative profitability of developing differenttechnologies? It is more profitable to develop technologies...

1 when the goods produced by these technologies command higher prices(price effect);

2 that have a larger market (market size effect).



Equilibrium Relative Bias

Potentially counteracting effects, but the market size effect will bemore powerful often.

Under fairly general conditions:

Weak Equilibrium (Relative) Bias: an increase in the relative supply ofa factor always induces technological change that is biased in favor ofthis factor.Strong Equilibrium (Relative) Bias: if the elasticity of substitutionbetween factors is suffi ciently large, an increase in the relative supply ofa factor induces suffi ciently strong technological change biased towardsitself that the endogenous-technology relative demand curve of theeconomy becomes upward-sloping.



Equilibrium Relative Bias in More Detail I

Suppose the (inverse) relative demand curve:

wH/wL = D (H/L,A)

where wH/wL is the relative price of the factors and A is a technologyterm.

A is H-biased if D is increasing in A, so that a higher A increases therelative demand for the H factor.

D is always decreasing in H/L.Equilibrium bias: behavior of A as H/L changes,

A (H/L)



Equilibrium Relative Bias in More Detail II

Weak equilibrium bias:

A (H/L) is increasing (nondecreasing) in H/L.

Strong equilibrium bias:

A (H/L) is suffi ciently responsive to an increase in H/L that the totaleffect of the change in relative supply H/L is to increase wH/wL.i.e., let the endogenous-technology relative demand curve be

wH/wL = D (H/L,A (H/L)) ≡ D (H/L)

→Strong equilibrium bias: D increasing in H/L.


Biased Technological Change Basics and Definitions

Factor-augmenting technological change

Production side of the economy:

Y (t) = F (L (t) ,H (t) ,A (t)) ,

where ∂F/∂A > 0.

Technological change is L-augmenting if

∂F (L,H,A)∂A

≡ LA

∂F (L,H,A)∂L

.

Equivalent to:

the production function taking the special form, F (AL,H).Harrod-neutral technological change when L corresponds to labor andH to capital.

H-augmenting defined similarly, and corresponds to F (L,AH).



Factor-biased technological change

Technological change change is L-biased, if:

∂∂F (L,H ,A)/∂L∂F (L,H ,A)/∂H

∂A≥ 0.

Skill premiumRelative supplyof skills

H/L

Skillbiased tech. change

ω

ω’

Relative demandfor skills

Figure: The effect of H-biased technological change on relative demand andrelative factor prices.



Constant Elasticity of Substitution Production Function I

CES production function case:

Y (t) =[γL (AL (t) L (t))

σ−1σ + γH (AH (t)H (t))

σ−1σ

] σσ−1,

whereAL (t) and AH (t) are two separate technology terms.γi s determine the importance of the two factors, γL + γH = 1.σ ∈ (0,∞)=elasticity of substitution between the two factors.

σ = ∞, perfect substitutes, linear production function is linear.σ = 1, Cobb-Douglas,σ = 0, no substitution, Leontieff.σ > 1, “gross substitutes,”σ < 1, “gross complements”.

Clearly, AL (t) is L-augmenting, while AH (t) is H-augmenting.Whether technological change that is L-augmenting (orH-augmenting) is L-biased or H-biased depends on σ.



Constant Elasticity of Substitution Production Function II

Relative marginal product of the two factors:

MPHMPL

= γ

(AH (t)AL (t)

) σ−1σ(H (t)L (t)

)− 1σ

, (1)

where γ ≡ γH/γL.substitution effect: the relative marginal product of H is decreasing inits relative abundance, H (t) /L (t).The effect of AH (t) on the relative marginal product:

If σ > 1, an increase in AH (t) (relative to AL (t)) increases therelative marginal product of H.If σ < 1, an increase in AH (t) reduces the relative marginal product ofH.If σ = 1, Cobb-Douglas case, and neither a change in AH (t) nor inAL (t) is biased towards any of the factors.

Note also that σ is the elasticity of substitution between the twofactors.



Constant Elasticity of Substitution Production Function III

Intuition for why, when σ < 1, H-augmenting technical change isL-biased:

with gross complementarity (σ < 1), an increase in the productivity ofH increases the demand for labor, L, by more than the demand for H,creating “excess demand” for labor.the marginal product of labor increases by more than the marginalproduct of H.Take case where σ→ 0 (Leontieff): starting from a situation in whichγLAL (t) L (t) = γHAH (t)H (t), a small increase in AH (t) will createan excess of the services of the H factor, and its price will fall to 0.



Equilibrium Bias

Weak equilibrium bias of technology: an increase in H/L, inducestechnological change biased towards H. i.e., given (1):

d (AH (t) /AL (t))σ−1

σ

dH/L≥ 0,

so AH (t) /AL (t) is biased towards the factor that has become moreabundant.Strong equilibrium bias: an increase in H/L induces a suffi cientlylarge change in the bias so that the relative marginal product of Hrelative to that of L increases following the change in factor supplies:

dMPH/MPLdH/L

> 0,

The major difference is whether the relative marginal product of thetwo factors are evaluated at the initial relative supplies (weak bias) orat the new relative supplies (strong bias).


Baseline Model of Directed Technical Change Environment

Baseline Model of Directed Technical Change I

Framework: expanding varieties model with lab equipmentspecification of the innovation possibilities frontier (so none of theresults here depend on technological externalities).

Constant supply of L and H.

Representative household with the standard CRRA preferences:

∫ ∞

0exp (−ρt)

C (t)1−θ − 11− θ

dt, (2)

Aggregate production function:

Y (t) =[γLYL (t)

ε−1ε + γHYH (t)

ε−1ε

] εε−1, (3)

where intermediate good YL (t) is L-intensive, YH (t) is H-intensive.



Baseline Model of Directed Technical Change II

Resource constraint (define Z (t) = ZL (t) + ZH (t)):

C (t) + X (t) + Z (t) ≤ Y (t) , (4)

Intermediate goods produced competitively with:

YL (t) =1

1− β

(∫ NL(t)

0xL (ν, t)

1−β dν

)Lβ (5)

and

YH (t) =1

1− β

(∫ NH (t)

0xH (ν, t)

1−β dν

)Hβ, (6)

where machines xL (ν, t) and xH (ν, t) are assumed to depreciate afteruse.



Baseline Model of Directed Technical Change III

Differences with baseline expanding product varieties model:1 These are production functions for intermediate goods rather than thefinal good.

2 (5) and (6) use different types of machines—different ranges [0,NL (t)]and [0,NH (t)].

All machines are supplied by monopolists that have a fully-enforcedperpetual patent, at prices pxL (ν, t) for ν ∈ [0,NL (t)] and pxH (ν, t)for ν ∈ [0,NH (t)].Once invented, each machine can be produced at the fixed marginalcost ψ in terms of the final good.

Normalize to ψ ≡ 1− β.



Baseline Model of Directed Technical Change IV

Total resources devoted to machine production at time t are

X (t) = (1− β)

(∫ NL(t)

0xL (ν, t) dν+

∫ NH (t)

0xH (ν, t) dν

).

Innovation possibilities frontier:

NL (t) = ηLZL (t) and NH (t) = ηHZH (t) , (7)

Value of a monopolist that discovers one of these machines is:

Vf (ν, t) =∫ ∞

texp

[−∫ s

tr(s ′)ds ′]

πf (ν, s)ds, (8)

where πf (ν, t) ≡ pxf (ν, t)xf (ν, t)− ψxf (ν, t) for f = L or H.Hamilton-Jacobi-Bellman version:

r (t)Vf (ν, t)− Vf (ν, t) = πf (ν, t). (9)



Baseline Model of Directed Technical Change V

Normalize the price of the final good at every instant to 1, which isequivalent to setting the ideal price index of the two intermediatesequal to one, i.e.,[

γεL (pL (t))

1−ε + γεH (pH (t))

1−ε] 11−ε= 1 for all t, (10)

where pL (t) is the price index of YL at time t and pH (t) is the priceof YH .

Denote factor prices by wL (t) and wH (t).


Baseline Model of Directed Technical Change Characterization of Equilibrium

Equilibrium I

Allocation. Time paths of

[C (t) ,X (t) ,Z (t)]∞t=0,[NL (t) ,NH (t)]

∞t=0,[

pxL (ν, t) , xL (ν, t) ,VL (ν, t)]∞

t=0,ν∈[0,NL(t)]

and

[χH (ν, t) , xH (ν, t) ,VH (ν, t)]∞

t=0,ν∈[0,NH (t)]

, and

[r (t) ,wL (t) ,wH (t)]∞t=0.

Equilibrium. An allocation in which

All existing research firms choose[pxf (ν, t) , xf (ν, t)

]∞t=0,

ν∈[0,Nf (t)]for f = L, H to maximize profits,

[NL (t) ,NH (t)]∞t=0 is determined by free entry

[r (t) ,wL (t) ,wH (t)]∞t=0, are consistent with market clearing, and

[C (t) ,X (t) ,Z (t)]∞t=0 are consistent with consumer optimization.



Equilibrium II

Maximization problem of producers in the two sectors:

maxL,[xL(ν,t)]ν∈[0,NL (t)]

pL (t)YL (t)− wL (t) L (11)

−∫ NL(t)

0pxL (ν, t) xL (ν, t) dν,

and

maxH ,[xH (ν,t)]ν∈[0,NH (t)]

pH (t)YH (t)− wH (t)H (12)

−∫ NH (t)

0pxH (ν, t) xH (ν, t) dν.

Note the presence of pL (t) and pH (t), since these sectors produceintermediate goods.



Equilibrium III

Thus, demand for machines in the two sectors:

xL (ν, t) =[pL (t)pxL (ν, t)

]1/β

L for all ν ∈ [0,NL (t)] and all t, (13)

and

xH (ν, t) =[pH (t)pxH (ν, t)

]1/β

H for all ν ∈ [0,NH (t)] and all t. (14)

Maximization of the net present discounted value of profits implies aconstant markup:

pxL (ν, t) = pxH (ν, t) = 1 for all ν and t.



Equilibrium IV

Substituting into (13) and (14):

xL (ν, t) = pL (t)1/β L for all ν and all t,

andxH (ν, t) = pH (t)

1/β H for all ν and all t.

Since these quantities do not depend on the identity of the machineprofits are also independent of the machine type:

πL (t) = βpL (t)1/β L and πH (t) = βpH (t)

1/β H. (15)

Thus the values of monopolists only depend on which sector they are,VL (t) and VH (t).



Equilibrium V

Combining these with (5) and (6), derived production functions forthe two intermediate goods:

YL (t) =1

1− βpL (t)

1−ββ NL (t) L (16)

andYH (t) =

11− β

pH (t)1−β

β NH (t)H. (17)



Equilibrium VI

For the prices of the two intermediate goods, (3) imply

p (t) ≡ pH (t)pL (t)

= γ

(YH (t)YL (t)

)− 1ε

= γ

(p (t)

1−ββNH (t)HNL (t) L

)− 1ε

= γεβσ

(NH (t)HNL (t) L

)− βσ

, (18)

where γ ≡ γH/γL and

σ ≡ ε− (ε− 1) (1− β)

= 1+ (ε− 1) β.



Equilibrium VII

We can also calculate the relative factor prices:

ω (t) ≡ wH (t)wL (t)

= p (t)1/β NH (t)NL (t)

= γεσ

(NH (t)NL (t)

) σ−1σ(HL

)− 1σ

. (19)

σ is the (derived) elasticity of substitution between the two factors,since it is exactly equal to

σ = −(d logω (t)d log (H/L)

)−1.



Equilibrium VIII

Free entry conditions:

ηLVL (t) ≤ 1 and ηLVL (t) = 1 if ZL (t) > 0. (20)

andηHVH (t) ≤ 1 and ηHVH (t) = 1 if ZH (t) > 0. (21)

Consumer side:C (t)C (t)

=1θ(r (t)− ρ) , (22)

and

limt→∞

[exp

(−∫ t

0r (s) ds

)(NL (t)VL (t) +NH (t)VH (t))

]= 0,

(23)where NL (t)VL (t) +NH (t)VH (t) is the total value of corporateassets in this economy.



Balanced Growth Path I

Consumption grows at the constant rate, g ∗, and the relative pricep (t) is constant. From (10) this implies that pL (t) and pH (t) arealso constant.

Let VL and VH be the BGP net present discounted values of newinnovations in the two sectors. Then (9) implies that

VL =βp1/βL Lr ∗

and VH =βp1/βH Hr ∗

, (24)

Taking the ratio of these two expressions, we obtain

VHVL

=

(pHpL

) 1β HL.



Balanced Growth Path II

Note the two effects on the direction of technological change:1 The price effect: VH/VL is increasing in pH/pL. Tends to favortechnologies complementing scarce factors.

2 The market size effect: VH/VL is increasing in H/L. It encouragesinnovation for the more abundant factor.

The above discussion is incomplete since prices are endogenous.Combining (24) together with (18):

VHVL

=

(1− γ

γ

) εσ(NHNL

)− 1σ(HL

) σ−1σ

. (25)

Note that an increase in H/L will increase VH/VL as long as σ > 1and it will reduce it if σ < 1. Moreover,

σ T 1 ⇐⇒ ε T 1.The two factors will be gross substitutes when the two intermediategoods are gross substitutes in the production of the final good.



Balanced Growth Path III

Next, using the two free entry conditions (20) and (21) as equalities,we obtain the following BGP “technology market clearing” condition:

ηLVL = ηHVH . (26)

Combining this with (25), BGP ratio of relative technologies is(NHNL

)∗= ησγε

(HL

)σ−1, (27)

where η ≡ ηH/ηL.

Note that relative productivities are determined by the innovationpossibilities frontier and the relative supply of the two factors. In thissense, this model totally endogenizes technology.



Summary of Balanced Growth Path

Proposition Consider the directed technological change model describedabove. Suppose

β[γεH (ηHH)

σ−1 + γεL (ηLL)

σ−1] 1

σ−1> ρ(28)

and (1− θ) β[γεH (ηHH)


σ−1] 1

σ−1< ρ.

Then there exists a unique BGP equilibrium in which therelative technologies are given by (27), and consumption andoutput grow at the rate

g ∗ =1θ

(β[γεH (ηHH)


σ−1] 1

σ−1 − ρ

). (29)


Baseline Model of Directed Technical Change Transitional Dynamics

Transitional Dynamics

Differently from the baseline endogenous technological changemodels, there are now transitional dynamics (because there are twostate variables).

Nevertheless, transitional dynamics simple and intuitive:

Proposition Consider the directed technological change model describedabove. Starting with any NH (0) > 0 and NL (0) > 0, thereexists a unique equilibrium path. IfNH (0) /NL (0) < (NH/NL)

∗ as given by (27), then we haveZH (t) > 0 and ZL (t) = 0 untilNH (t) /NL (t) = (NH/NL)

∗. IfNH (0) /NL (0) < (NH/NL)

∗, then ZH (t) = 0 andZL (t) > 0 until NH (t) /NL (t) = (NH/NL)

∗.

Summary: the dynamic equilibrium path always tends to the BGP andduring transitional dynamics, there is only one type of innovation.


Baseline Model of Directed Technical Change Directed Technological Change and Factor Prices

Directed Technological Change and Factor Prices

In BGP, there is a positive relationship between H/L and N∗H/N∗Lonly when σ > 1.

But this does not mean that depending on σ (or ε), changes in factorsupplies may induce technological changes that are biased in favor oragainst the factor that is becoming more abundant.

Why?

N∗H/N∗L refers to the ratio of factor-augmenting technologies, or to theratio of physical productivities.What matters for the bias of technology is the value of marginalproduct of factors, affected by relative prices.The relationship between factor-augmenting and factor-biasedtechnologies is reversed when σ is less than 1.When σ > 1, an increase in N∗H/N∗L is relatively biased towards H,while when σ < 1, a decrease in N∗H/N∗L is relatively biased towards H.



Weak Equilibrium (Relative) Bias Result

Proposition Consider the directed technological change model describedabove. There is always weak equilibrium (relative) bias inthe sense that an increase in H/L always induces relativelyH-biased technological change.

The results reflect the strength of the market size effect: it alwaysdominates the price effect.

But it does not specify whether this induced effect will be strongenough to make the endogenous-technology relative demand curve forfactors upward-sloping.



Strong Equilibrium (Relative) Bias Result

Substitute for (NH/NL)∗ from (27) into the expression for the

relative wage given technologies, (19), and obtain:

ω∗ ≡(wHwL

)∗= ησ−1γε

(HL

)σ−2. (30)

Proposition Consider the directed technological change model describedabove. Then if σ > 2, there is strong equilibrium(relative) bias in the sense that an increase in H/L raisesthe relative marginal product and the relative wage of thefactor H compared to factor L.



Relative Supply of Skills and Skill Premium

Skill premium

Relative Supply of Skills

CTconstanttechnologydemand

ET1endogenoustechnologydemand

ET2endogenoustechnology demand



Discussion

Analogous to Samuelson’s LeChatelier principle: think of theendogenous-technology demand curve as adjusting the “factors ofproduction” corresponding to technology.

But, the effects here are caused by general equilibrium changes, noton partial equilibrium effects.

Moreover ET2, which applies when σ > 2 holds, is upward-sloping.

A complementary intuition: importance of non-rivalry of ideas:

leads to an aggregate production function that exhibits increasingreturns to scale (in all factors including technologies).the market size effect can create suffi ciently strong inducedtechnological change to increase the relative marginal product and therelative price of the factor that has become more abundant.


Baseline Model of Directed Technical Change Implications

Implications I

Recall we have the following stylized facts:

Secular skill-biased technological change increasing the demand forskills throughout the 20th century.Possible acceleration in skill-biased technological change over the past25 years.A range of important technologies biased against skill workers duringthe 19th century.

The current model gives us a way to think about these issues.

The increase in the number of skilled workers should cause steadyskill-biased technical change.Acceleration in the increase in the number of skilled workers shouldinduce an acceleration in skill-biased technological change.Available evidence suggests that there were large increases in thenumber of unskilled workers during the late 18th and 19th centuries.



Implications II

The framework also gives a potential interpretation for the dynamicsof the college premium during the 1970s and 1980s.

It is reasonable that the equilibrium skill bias of technologies, NH/NL,is a sluggish variable.Hence a rapid increase in the supply of skills would first reduce the skillpremium as the economy would be moving along a constant technology(constant NH/NL).After a while technology would start adjusting, and the economy wouldmove back to the upward sloping relative demand curve, with arelatively sharp increase in the college premium.



Implications III

Skill premium

Longrun relativedemand for skills

Exogenous Shift inRelative Supply

Initial premium

ShortrunResponse

Longrun premium

Figure: Dynamics of the skill premium in response to an exogenous increase inthe relative supply of skills, with an upward-sloping endogenous-technologyrelative demand curve.



Implications IV

If instead σ < 2, the long-run relative demand curve will be downwardsloping, though again it will be shallower than the short-run relativedemand curve.

An increase in the relative supply of skills leads again to a decline inthe college premium, and as technology starts adjusting the skillpremium will increase.

But it will end up below its initial level. To explain the larger increasein the college premium in the 1980s, in this case we would need someexogenous skill-biased technical change.



Implications V

Skill premium

Longrun relativedemand for skills

Exogenous Shift inRelative Supply

Initial premium

ShortrunResponse

Longrun premium

Figure: Dynamics of the skill premium in response to an increase in the relativesupply of skills, with a downward-sloping endogenous-technology relative demandcurve.



Implications VI

Other remarks:

Upward-sloping relative demand curves arise only when σ > 2. Mostestimates put the elasticity of substitution between 1.4 and 2. Onewould like to understand whether σ > 2 is a feature of the specificmodel discussed hereResults on induced technological change are not an artifact of the scaleeffect (exactly the same results apply when scale effects are removed,see below).


Baseline Model of Directed Technical Change Pareto Optimal Allocations

Pareto Optimal Allocations I

The social planner would not charge a markup on machines:

xSL (ν, t) = (1− β)−1/β pL (t)1/β L

and xSH (ν, t) = (1− β)−1/β pH (t)1/β H.

Thus:

Y S (t) = (1− β)−1/β β[γεL

(NSL (t) L

) σ−1σ

(31)

+γεH

(NSH (t)H

) σ−1σ]

σσ−1 .



Pareto Optimal Allocations II

The current-value Hamiltonian is:

H (·) =CS (t)1−θ − 1

1− θ

+µL (t) ηLZSL (t) + µH (t) ηHZ

SH (t) ,

subject to

CS (t) = (1− β)−1/β[

γεL

(NSL (t) L

) σ−1σ+ γε

H

(NSH (t)H

) σ−1σ

] σσ−1

−ZSL (t)− ZSH (t) .



Summary of Pareto Optimal Allocations

Proposition The stationary solution of the Pareto optimal allocationinvolves relative technologies given by (27) as in thedecentralized equilibrium. The stationary growth rate ishigher than the equilibrium growth rate and is given by

gS =1θ

((1− β)−1/β β

[(1− γ)ε (ηHH)

σ−1 + γε (ηLL)σ−1] 1

σ−1 − ρ

)> g ∗,

where g ∗ is the BGP growth rate given in (29).


Directed Technological Change with Knowledge Spillovers Environment

Directed Technological Change with Knowledge Spillovers I

The lab equipment specification of the innovation possibilities doesnot allow for state dependence.

Assume that R&D is carried out by scientists and that there is aconstant supply of scientists equal to S

With only one sector, sustained endogenous growth requires N/N tobe proportional to S .

With two sectors, there is a variety of specifications with differentdegrees of state dependence, because productivity in each sector candepend on the state of knowledge in both sectors.

A flexible formulation is

NL (t) = ηLNL (t)(1+δ)/2 NH (t)

(1−δ)/2 SL (t) (32)

and NH (t) = ηHNL (t)(1−δ)/2 NH (t)

(1+δ)/2 SH (t) ,

where δ ≤ 1.Daron Acemoglu (MIT) Economic Growth Lecture 13 December 13, 2011. 50 / 71


Directed Technological Change with Knowledge SpilloversII

Market clearing for scientists requires that

SL (t) + SH (t) ≤ S . (33)

δ measures the degree of state-dependence:δ = 0. Results are unchanged. No state-dependence:(

∂NH/∂SH)

/(∂NL/∂SL

)= ηH/ηL

irrespective of the levels of NL and NH .Both NL and NH create spillovers for current research in both sectors.δ = 1. Extreme amount of state-dependence:(

∂NH/∂SH)

/(∂NL/∂SL

)= ηHNH/ηLNL

an increase in the stock of L-augmenting machines today makes futurelabor-complementary innovations cheaper, but has no effect on thecost of H-augmenting innovations.



Directed Technological Change with Knowledge SpilloversIII

State dependence adds another layer of “increasing returns,” this timenot for the entire economy, but for specific technology lines.

Free entry conditions:

ηLNL (t)(1+δ)/2 NH (t)

(1−δ)/2 VL (t) ≤ wS (t) (34)

and ηLNL (t)(1+δ)/2 NH (t)

(1−δ)/2 VL (t) = wS (t) if SL (t) > 0.

and

ηHNL (t)(1−δ)/2 NH (t)

(1+δ)/2 VH (t) ≤ wS (t) (35)

and ηHNL (t)(1−δ)/2 NH (t)

(1+δ)/2 VH (t) = wS (t) if SH (t) > 0,

where wS (t) denotes the wage of a scientist at time t.



Directed Technological Change with Knowledge SpilloversIV

When both of these free entry conditions hold, BGP technologymarket clearing implies

ηLNL (t)δ πL = ηHNH (t)

δ πH , (36)

Combine condition (36) with equations (15) and (18), to obtain theequilibrium relative technology as:(

NHNL

)∗= η

σ1−δσ γ

ε1−δσ

(HL

) σ−11−δσ

, (37)

where γ ≡ γH/γL and η ≡ ηH/ηL.



Directed Technological Change with Knowledge SpilloversV

The relationship between the relative factor supplies and relativephysical productivities now depends on δ.

This is intuitive: as long as δ > 0, an increase in NH reduces therelative costs of H-augmenting innovations, so for technology marketequilibrium to be restored, πL needs to fall relative to πH .

Substituting (37) into the expression (19) for relative factor prices forgiven technologies, yields the following long-run(endogenous-technology) relationship:

ω∗ ≡(wHwL

)∗= η

σ−11−δσ γ

(1−δ)ε1−δσ

(HL

) σ−2+δ1−δσ

. (38)



Directed Technological Change with Knowledge SpilloversVI

The growth rate is determined by the number of scientists. In BGPwe need NL (t) /NL (t) = NH (t) /NH (t), or

ηHNH (t)δ−1 SH (t) = ηLNL (t)

δ−1 SL (t) .

Combining with (33) and (37), BGP allocation of researchers betweenthe two different types of technologies:

η1−σ1−δσ

(1− γ

γ

)− ε(1−δ)1−δσ

(HL

)− (σ−1)(1−δ)1−δσ

=S∗L

S − S∗L, (39)

Notice that given H/L, the BGP researcher allocations, S∗L and S∗H ,

are uniquely determined.


Directed Technological Change with Knowledge Spillovers Balanced Growth Path

Balanced Growth Path with Knowledge Spillovers

Proposition Consider the directed technological change model withknowledge spillovers and state dependence in the innovationpossibilities frontier. Suppose that

(1− θ)ηLηH (NH/NL)

(δ−1)/2

ηH (NH/NL)(δ−1) + ηL

S < ρ,

where NH/NL is given by (37). Then there exists a uniqueBGP equilibrium in which the relative technologies are givenby (37), and consumption and output grow at the rate

g ∗ =ηLηH (NH/NL)

(δ−1)/2

ηH (NH/NL)(δ−1) + ηL

S . (40)


Directed Technological Change with Knowledge Spillovers Transitional Dynamics

Transitional Dynamics with Knowledge Spillovers

Transitional dynamics now more complicated because of the spillovers.

The dynamic equilibrium path does not always tend to the BGPbecause of the additional increasing returns to scale:

With a high degree of state dependence, when NH (0) is very highrelative to NL (0), it may no longer be profitable for firms to undertakefurther R&D directed at labor-augmenting (L-augmenting)technologies.Whether this is so or not depends on a comparison of the degree ofstate dependence, δ, and the elasticity of substitution, σ.



Summary of Transitional Dynamics

Proposition Suppose thatσ < 1/δ.

Then, starting with any NH (0) > 0 and NL (0) > 0, thereexists a unique equilibrium path. IfNH (0) /NL (0) < (NH/NL)

∗ as given by (37), then we haveZH (t) > 0 and ZL (t) = 0 untilNH (t) /NL (t) = (NH/NL)

∗. NH (0) /NL (0) < (NH/NL)∗,

then ZH (t) = 0 and ZL (t) > 0 untilNH (t) /NL (t) = (NH/NL)

∗.If

σ > 1/δ,

then starting with NH (0) /NL (0) > (NH/NL)∗, the

economy tends to NH (t) /NL (t)→ ∞ as t → ∞, andstarting with NH (0) /NL (0) < (NH/NL)

∗, it tends toNH (t) /NL (t)→ 0 as t → ∞.



Equilibrium Relative Bias with Knowledge Spillovers I

Proposition Consider the directed technological change model withknowledge spillovers and state dependence in the innovationpossibilities frontier. Then there is always weak equilibrium(relative) bias in the sense that an increase in H/L alwaysinduces relatively H-biased technological change.

Proposition Consider the directed technological change model withknowledge spillovers and state dependence in the innovationpossibilities frontier. Then if

σ > 2− δ,

there is strong equilibrium (relative) bias in the sense thatan increase in H/L raises the relative marginal product andthe relative wage of the H factor compared to the L factor.



Equilibrium Relative Bias with Knowledge Spillovers II

Intuitively, the additional increasing returns to scale coming fromstate dependence makes strong bias easier to obtain, because theinduced technology effect is stronger.

Note the elasticity of substitution between skilled and unskilled laborsignificantly less than 2 may be suffi cient to generate strongequilibrium bias.

How much lower than 2 the elasticity of substitution can be dependson the parameter δ. Unfortunately, this parameter is not easy tomeasure in practice.


Endogenous Labor-Augmenting Technological Change Labor-Augmenting Change

Endogenous Labor-Augmenting Technological Change I

Models of directed technological change create a natural reason fortechnology to be more labor augmenting than capital augmenting.

Under most circumstances, the resulting equilibrium is not purelylabor augmenting and as a result, a BGP fails to exist.

But in one important special case, the model delivers long-run purelylabor augmenting technological changes exactly as in the neoclassicalgrowth model.

Consider a two-factor model with H corresponding to capital, that is,H (t) = K (t).

Assume that there is no depreciation of capital.

Note that in this case the price of the second factor, K (t), is thesame as the interest rate, r (t).

Empirical evidence suggests σ < 1 and is also economically plausible.



Endogenous Labor-Augmenting Technological Change II

Recall that when σ < 1 labor-augmenting technological changecorresponds to capital-biased technological change.

Hence the questions are:1 Under what circumstances would the economy generate relativelycapital-biased technological change?

2 When will the equilibrium technology be suffi ciently capital biased thatit corresponds to Harrod-neutral technological change?



Endogenous Labor-Augmenting Technological Change III

To answer 1, note that what distinguishes capital from labor is thefact that it accumulates.

The neoclassical growth model with technological change experiencescontinuous capital-deepening as K (t) /L increases.This implies that technological change should be morelabor-augmenting than capital augmenting.

Proposition In the baseline model of directed technological change withH (t) = K (t) as capital, if K (t) /L is increasing over timeand σ < 1, then NL (t) /NK (t) will also increase over time.



Endogenous Labor-Augmenting Technological Change IV

But the results are not easy to reconcile with purely-labor augmentingtechnological change. Suppose that capital accumulates at anexogenous rate, i.e.,

K (t)K (t)

= sK > 0. (41)

Proposition Consider the baseline model of directed technological changewith the knowledge spillovers specification and statedependence. Suppose that δ < 1 and capital accumulatesaccording to (41). Then there exists no BGP.

Intuitively, even though technological change is more laboraugmenting than capital augmenting, there is still capital-augmentingtechnological change in equilibrium.Moreover it can be proved that in any asymptotic equilibrium, r (t)cannot be constant, thus consumption and output growth cannot beconstant.



Endogenous Labor-Augmenting Technological Change V

Special case that justifies the basic structure of the neoclassicalgrowth model: extreme state dependence (δ = 1).

In this case:r (t)K (t)wL (t) L

= η−1. (42)

Thus, directed technological change ensures that the share of capitalis constant in national income. .

Recall from (19) that

r (t)wL (t)

= γεσ

(NK (t)NL (t)

) σ−1σ(K (t)L

)− 1σ

,

where γ ≡ γK/γL and γK replaces γH in the production function(3).



Endogenous Labor-Augmenting Technological Change VI

Consequently,

r (t)K (t)wL (t) L (t)

= γεσ

(NK (t)NL (t)

) σ−1σ(K (t)L

) σ−1σ

.

In this case, (42) combined with (41) implies that

NL (t)NL (t)

− NK (t)NK (t)

= sK . (43)

Moreover:

r (t) = βγKNK (t)

[γL

(NL (t) L

NK (t)K (t)

) σ−1σ

+ γL)

] 1σ−1

. (44)



Endogenous Labor-Augmenting Technological Change VII

From (22), a constant growth path which consumption grows at aconstant rate is only possible if r (t) is constant.

Equation (43) implies that (NL (t) L) / (NK (t)K (t)) is constant,thus NK (t) must also be constant.

Therefore, equation (43) implies that technological change must bepurely labor augmenting.



Summary of Endogenous Labor-Augmenting TechnologicalChange

Proposition Consider the baseline model of directed technological changewith the two factors corresponding to labor and capital.Suppose that the innovation possibilities frontier is given bythe knowledge spillovers specification and extreme statedependence, i.e., δ = 1 and that capital accumulatesaccording to (41). Then there exists a constant growth pathallocation in which there is only labor-augmentingtechnological change, the interest rate is constant andconsumption and output grow at constant rates. Moreover,there cannot be any other constant growth path allocations.



Stability

The constant growth path allocation with purely labor augmentingtechnological change is globally stable if σ < 1.Intuition:

If capital and labor were gross substitutes (σ > 1), the equilibriumwould involve rapid accumulation of capital and capital-augmentingtechnological change, leading to an asymptotically increasing growthrate of consumption.When capital and labor are gross complements (σ < 1), capitalaccumulation would increase the price of labor and profits fromlabor-augmenting technologies and thus encourage furtherlabor-augmenting technological change.σ < 1 forces the economy to strive towards a balanced allocation ofeffective capital and labor units.Since capital accumulates at a constant rate, a balanced allocationimplies that the productivity of labor should increase faster, and theeconomy should converge to an equilibrium path with purelylabor-augmenting technological progress.


Conclusions Conclusions

Conclusions I

The bias of technological change is potentially important for thedistributional consequences of the introduction of new technologies(i.e., who will be the losers and winners?); important for politicaleconomy of growth.

Models of directed technological change enable us to investigate arange of new questions:

the sources of skill-biased technological change over the past 100 years,the causes of acceleration in skill-biased technological change duringmore recent decades,the causes of unskilled-biased technological developments during the19th century,the relationship between labor market institutions and the types oftechnologies that are developed and adopted,why technological change in neoclassical-type models may be largelylabor-augmenting.


Conclusions Conclusions

Conclusions II

The implications of the class of models studied for the empiricalquestions mentioned above stem from the weak equilibrium bias andstrong equilibrium bias results.

Technology should not be thought of as a black box. Profit incentiveswill play a major role in both the aggregate rate of technologicalprogress and also in the biases of the technologies.


Date post:	12-Sep-2021
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

14.452 Economic Growth: Lecture 13, Directed Technological Change

Documents